1. Field of the Invention
The present invention relates generally to wire-duct electrostatic precipitators, and particularly to a method of modeling fly ash collection efficiency in wire-duct electrostatic precipitators.
2. Description of the Related Art
An electrostatic precipitator (ESP), or electrostatic air cleaner, is a particulate collection device that removes particles from a flowing gas (such as air) using the force of an induced electrostatic charge. Electrostatic precipitators are highly efficient filtration devices that minimally impede the flow of gases through the device, and can easily remove fine particulate matter, such as dust and smoke, from the air stream. In contrast to wet scrubbers, which apply energy directly to the flowing fluid medium, an ESP applies energy only to the particulate matter being collected, and therefore is very efficient in its consumption of energy (in the form of electricity).
The most basic precipitator contains a row of thin vertical wires, followed by a stack of large flat metal plates oriented vertically. The plates are typically spaced about 1 cm to 18 cm apart, depending on the particular application. The air or gas stream flows horizontally through the spaces between the wires, and then passes through the stack of plates. A negative voltage of several thousand volts is applied between the wires and plates. If the applied voltage is high enough, an electric (corona) discharge ionizes the gas around the electrodes. Negative ions flow to the plates and charge the gas-flow particles. The ionized particles, following the negative electric field created by the power supply, move to the grounded plates. Particles build up on the collection plates and form a layer. The layer does not collapse, due to electrostatic pressure (given from layer resistivity, electric field, and current flowing in the collected layer).
FIG. 1 diagrammatically illustrates a wire-duct electrostatic precipitator (WDEP) 100, and FIG. 2 is a schematic diagram of the WDEP of FIG. 1, illustrating some parameters of interest. A high voltage source HV is connected to high voltage rods 102, which have discharge wires 104 extending therebetween. Conductive plates 106 are placed on either side of the rods 102, and each plate 106 is grounded.
When the applied voltage is raised, the gas near the more sharply curved wire electrodes 104 breaks down at a voltage above what is referred to as the “onset value” and less than the “spark breakdown value”. This incomplete dielectric breakdown, which is called a “monopolar corona”, appears in air as a highly active region of glow. The monopolar corona within duct-type precipitators includes only positive or negative ions (the back corona is neglected), the polarity of the ions being the same as the polarity of the high voltage wires 104 in the corona. In FIGS. 1 and 2, the radius of each wire 104 is represented as R; S represents the wire-to-plate spacing (i.e., the distance between wires 104 and one of plates 106, measured along the Y-axis, as shown in FIG. 2); D represents the wire-to-wire spacing; and H represents the precipitator length (i.e., the length of each plate 106 measured along the X-axis, shown in FIG. 2).
For this configuration of WDEP, the following system of equations describes the monopolar corona:∇·{right arrow over (E)}=ρ/∈0  (1)∇·{right arrow over (J)}=0  (2){right arrow over (E)}=−∇φ  (3){right arrow over (J)}={right arrow over (J)}io+{right arrow over (J)}p  (4){right arrow over (J)}io=kioρio{right arrow over (E)}  (5){right arrow over (J)}p=kpρp{right arrow over (E)}  (6)where {right arrow over (E)} is the electric field intensity vector, ρ is the total space charge density (i.e., the summation of the ion charge density ρio and the particle charge density ρp, or ρ=ρio+ρp), {right arrow over (J)} is the total current density vector, φ is the potential, ∈0 is the permittivity of free space, and kio and kp are the mobilities for ions and particles, respectively.
Equations (1)-(6) represent Poisson's equation, the current continuity equation, the field and potential relations, the total current density equation, and the ion and particle current density equations, respectively. The exact analytical solution to these equations can only be obtained for parallel plates, coaxial cylinders, and concentric spheres. Because of the nature of this problem, a numerical solution would be desirable to provide solutions for this set of equations.
The following assumptions and boundary conditions are essential requirements for finding a numerical solution: First, the influence of particle space charge density on the field may be approximated by assuming that the particle concentration Np is constant over a given cross section of the precipitator 106. The particle's specific surface Sp (i.e., the surface per unit volume of gas) is given as:Sp=4Πa2Np  (7)where a is the radius of assumed spherical particles.
The corona discharge is assumed to be distributed uniformly over the surface of the wires 104; if the corona electrode has a potential above a certain value (i.e., the onset level), the normal component of the electric field remains constant at the onset value E0. Second, the ion mobility is assumed to be constant. And third, the ion diffusion is ignored.
With regard to boundary conditions, the potential at the two plates 106 is considered to be zero. Further, the potential at the discharging wires 104 is the potential of the source HV, which is denoted as V in the following. Lastly, the electric field at the discharging wires is E0, which is given by:
                              E          0                =                  3.1          ×                      10            6                    ⁢                                    (                              1                +                                  0.308                                                            0.5                      ×                      R                                                                                  )                        .                                              (        8        )            
Due to the complexities involved in the construction of solutions for equations (1)-(6), it is extremely difficult to optimize the collection of particulates, such as fly ash. However, it is often necessary to remove as many impurities from a fluid stream as possible. Thus, a method of modeling fly ash collection efficiency in wire-duct electrostatic precipitators solving the aforementioned problems is desired.